#help-26
topaz sinewBOT
frail ingot
Im confused how this work
mint crescent
The numerator can be rewritten as $$2^{17}+2^{17+1}+2^{17+2}=2^{17}+2^1 \cdot 2^{17}+2^2 \cdot 2^{17}$$ Factor out $2^{17}$ and you get the given thing
thorny flameBOT
Civil Service Pigeon
mint crescent
@frail ingot ^
mint crescent
we're using exponent rules to rewrite the numerator if that's what you're asking
frail ingot
Wait like when u wanna make 2^19 to 2^17. My teachet told me to do it like 2^17.2^2
mint crescent
Wait like when u wanna make 2^19 to 2^17. My teachet told me to do it like 2^17....
that's
frail ingot
Oooh
mint crescent
the same thing
mint crescent
you have 1 minute before I head off so make it quick
mint crescent
$\frac{2^{17}}{2^{18}}=2^{17-18}=2^{-1}=\frac{1}{2}$
thorny flameBOT
Civil Service Pigeon
frail ingot
Ooooooh
Boy im actually so dumb
I forgot that 1/2 existed
Thats about it
.close
topaz sinewBOT
topaz sinewBOT
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fast geode
topaz sinewBOT
rigid cloak
Bro break the logs fr
Like u can write log(3^2/4^2)
As log(3^2)-log(4^2)
It's log properties
Log(a/b)=loga -logb
And log(ab)=log a+logb
fast geode
oh rightttt right
fast geode
wait wait thanks thanks
topaz sinewBOT
@fast geode Has your question been resolved?
topaz sinewBOT
@fast geode Has your question been resolved?
swift copper
how to differentiate log a^x
is that logₐ(x)
topaz sinewBOT
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sharp dew
try to find the extremes of sin(2x) / 2+cos(2x)
rocky iron
try to find the extremes of sin(2x) / 2+cos(2x)
find the derivative using the quotient rule
finite storm
use chain rule and quotient rule
empty cloak
sharp dew
use chain rule and quotient rule
that's fked up
finite storm
):
empty cloak
not really
sharp dew
not a beautiful solution
finite storm
real
u can do that
or do logarithmic differentiation and chain rule
but u'd prefer the quotient rule if u want to find the extreme
sharp dew
rocky iron
i do not know how
you don't know trig differentiation or you don't know the quotient rule?
sharp dew
I differentiate it, and got the result
I do not even know whether it is correct or not
sharp dew
What
sharp dew
Can someone check it
finite storm
its ${\frac{(2+\cos(2x)\frac{d}{dx}(\sin(2x)) - \sin(2x)\frac{d}{dx}(2+\cos(2x)}{(2+\cos(2x)^2}}$
thorny flameBOT
:D
cedar wagon
You can say that its pi/2- periodic and an odd function so the interval of study can be reduce from R to
[-pi/3,pi/3]
sharp dew
oh, big brain huh
cedar wagon
Dont develop the denominator while quotient rule since we want to study the sign its better to let it as squared
sharp dew
You can say that its pi/2- periodic and an odd function so the interval of study...
isn't the period of the function pi?
sharp dew
Dont develop the denominator while quotient rule since we want to study the sign...
got it
ohhhh
rocky iron
only error is that you subtracted the numerator wrong way round, you did uv' - vu'
sharp dew
only error is that you subtracted the numerator wrong way round, you did uv' - v...
I see
rocky iron
it should be vu' - uv'
sharp dew
the quotient rule is a true nightmare
sharp dew
You can say that its pi/2- periodic and an odd function so the interval of study...
can someone explain how it works?
does it means I do not have to differtiate it at all
cedar wagon
does it means I do not have to differtiate it at all
No u have to
Its just a trig tips to not have make a sign table on R, and reduce it to a Lil interval which is way more efficient
cedar wagon
can someone explain how it works?
Pi/2 periodic leads to an interval of "length" pi/2 and the odd caracteristic allowed us to center make 0 the "center" of the interval
So [-pi/3,pi/3]
cedar wagon
It is
cedar wagon
No, it dont xd
sharp dew
the extreme occurs at -1/2
cos(2x) = -1/2
2x= 150 degree or 210 degree
x = 75 or 105
sharp dew
is it a convention of Yakuza
cedar wagon
Cos(2x) > -1/2
sharp dew
it is not an inequality
I mean cos(2x)=-1/2 is where the extreme occurs.
it is not an inequality.
cedar wagon
I mean cos(2x)=-1/2 is where the extreme occurs.
You dont have to precise if its minimum or maximum ?
sharp dew
can't I just put x= 5/6 pi and 7/6 pi respectively into f(x)
And compare the results.
i mean we find that 5/6 pi and 7/6 pi are of the points where the extremes occur
why not just put them into f(x) and compare their outputs then we are able to distinguish the max and min.
that's more convinient, no?
what we need inequality for
cedar wagon
You have to find with derivative not plugging some values, it does not prove
sharp dew
You have to find with derivative not plugging some values, it does not prove
it is not just plugging values, it is solution by logic.
it does prove by the way
cedar wagon
can't I just put x= 5/6 pi and 7/6 pi respectively into f(x)
How do you get these btw ?
sharp dew
the derivative signifies that x=5/6 pi and 7/6 pi we can get the extreme of f(x)
cedar wagon
the derivative signifies that x=5/6 pi and 7/6 pi we can get the extreme of f(x)
Are you sure of the result of x ?
It should be -pi/3 and pi/3
sharp dew
Given cos2x= -1/2 or -2 results in f'(x)=0
cedar wagon
Yeah so 2x = 2pi/3 or -2pi/3
sharp dew
That is, 2x= 2/3 pi and -2pi/3
I'm stress
to express angles with radian
Hence, x=1/3 pi and -1/3 pi is where the extremes occur
then
then
you could just plug in these two values into f(x)
and compare them
cedar wagon
Yeah ok
sharp dew
This could gain you the min and max.
topaz sinewBOT
@sharp dew Has your question been resolved?
glass kayak
How do I flatten a 2d array?
topaz sinewBOT
How do I flatten a 2d array?
opal vault
Just use the flatten function with numpy?
glass kayak
Not nesseceraly
opal vault
Not nesseceraly
Wdym
glass kayak
Just use the flatten function with numpy?
Sorry, miss read it
opal vault
numpy.ndarray.flatten(array)?
glass kayak
I mean, how do I flatten any 2d array, not neccesserekat in python
opal vault
Depends on the coding language then
glass kayak
Okay, how does numpy do it?
opal vault
If you have a matrix of coefficients $(a_{ij})_{1\leq i\leq n,1\leq j\leq m}$
thorny flameBOT
rafilou2003
opal vault
Then you create $(u_k)_{1\leq k \leq mn}$ by
thorny flameBOT
rafilou2003
opal vault
$u_{(m-1)i + j} = a_{ij}$
thorny flameBOT
rafilou2003
glass kayak
I'm sorry, I'm confused
opal vault
Tell me what you don’t understand
glass kayak
What's coefficents?
opal vault
What's coefficents?
The entries of the matrix
You know, a_ij means the entry on row i, column j
glass kayak
**rafilou2003**
So why do you need so many conditions?
opal vault
That's the length and width of the matrix
glass kayak
Oh okay
glass kayak
**rafilou2003**
Is it (m-1) * (i+j)?
topaz sinewBOT
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obtuse nest
are all boundary points limit points
topaz sinewBOT
are all boundary points limit points
topaz sinewBOT
@obtuse nest Has your question been resolved?
trim sail
Can someone help me out
topaz sinewBOT
Can someone help me out
fallow heart
I'm getting infinity - to infinity
Nope, are you sure? 🤔
fallow heart
I don't see any -∞
finite storm
the answer is infinity, right?
fallow heart
I don't see any -∞
+∞ yes but not negative ∞
fallow heart
Then why did you write ∞ - ∞ ? 😅
trim sail
I meant infinity to infinity
finite storm
like infty/infty?
trim sail
The limit is between ♾️<the function< ♾️
finite storm
ye
trim sail
That's what I'm unsure of
finite storm
cuz the limit to this is infinite
finite storm
yes
firstly, the dominant's term in the top is n^2 and the bottom n
the top grows faster than bottom so it goes to infty
trim sail
Yeah
finite storm
no
it oscillates between 0 and 1
but doesn't matter in comparison the size of infty (n^2)
trim sail
Yeah yeah it oscillates between 0 and 1 and continues that way to infinity
finite storm
yes
topaz sinewBOT
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indigo ice
using direct sub i get a very ugly number so im not sure if its right
mint crescent
using direct sub i get a very ugly number so im not sure if its right
What did you get
acoustic pecan
thats what they did
rigid cloak
,w limit x tends to -3 ((x-4)/(6x^2+2))^1/3
indigo ice
huh
thorny flameBOT
rigid cloak
-1/2
indigo ice
i got a -half
rigid cloak
Good
indigo ice
fair
indigo ice
.close
topaz sinewBOT
topaz sinewBOT
@plain glacier Has your question been resolved?
topaz sinewBOT
@plain glacier Has your question been resolved?
topaz sinewBOT
@plain glacier Has your question been resolved?
brave coral
so I was right 😏
anyhow, the last question seems pretty straightforward
Use the same idea from part b to find thee work done in moving an electron h metres through the field
Find the total number of electrons (n times length of wire)
you can probably figure out the last thing
brave coral
I think that could for very small h
although, h is the distance travelled (across wire i believe) not its displacement in the field
jaunty dawn
fallow heart
It's already reopened
topaz sinewBOT
@plain glacier Has your question been resolved?
topaz sinewBOT
@plain glacier Has your question been resolved?
indigo ice
im getting does not exist for a limit
topaz sinewBOT
im getting does not exist for a limit
indigo ice
not sure
noble laurel
You could recognize that sin(x)/cos(x) = tan(x)
And use your knowledge about its graph
noble laurel
Not always
For example substituting the x value here gives -1/0 which is undefined
but also there are cases, like functions with jump discontinuities, where the limit is not equal to the function evaluated at the limiting point
indigo ice
yeah i know about that
so perhaps i do tan(x)
and sub in the x?
that gets rid of the -1/0
noble laurel
tan(x) is undefined here
indigo ice
still?
noble laurel
Okay that's fine. Try instead taking the limit from the left hand side, and comparing it to the limit from the right hand side.
indigo ice
and pre cal b we didn't even touch the tan graphs
i haven't reached the left and right hand limits yet
i dont know how to get either
noble laurel
Well for the left hand limit for example
what you do is substitute values approaching 3pi/2, from the left
so try like 3pi/2-1, 3pi/2-0.1, 3pi/2-0.01, etc
and see what it approaches
indigo ice
okay
noble laurel
For the right hand limit, do the same thing but from the right hand side
indigo ice
and then the opposite for right
noble laurel
yes
noble laurel
And, the limit cannot exist if the left and right hand limits do not equal each other
indigo ice
And, the limit cannot exist if the left and right hand limits do not equal each ...
ok so
when its approaching from the left i get + values
noble laurel
yes
indigo ice
from the right its the same values but negative
noble laurel
yes
indigo ice
so im guessing they're approaching different infinity
noble laurel
You'd notice if you kept substituting that the limit from the LHS is positive infinity, and from the RHS is negative infinity
yes
indigo ice
so no limit?
noble laurel
,w plot y=tan(x), x=(3pi)/(2)
indigo ice
as left and right dont exist
thorny flameBOT
indigo ice
that looks about right
i only know cos and sin graphs haha
they skipped tan in pre cal a because we didnt have time
thanks tho
.close
topaz sinewBOT
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cloud gazelle
I'm really lost on this one. My problem is g(t) = sin(t^2 * cot(7t)). I'm supposed to find dg/dt. Any ideas? Is this just the chain rule?
topaz sinewBOT
I'm really lost on this one. My problem is g(t) = sin(t^2 * cot(7t)). I'm suppos...
raven sparrow
Yes. Chain rule.
Although you'll need the product rule to compute the inside derivative as well
topaz sinewBOT
@cloud gazelle Has your question been resolved?
cloud gazelle
gotcha, thanks!
hard egret
Need help solving for X.
topaz sinewBOT
Need help solving for X.
hard egret
<@&286206848099549185>
glacial cliff
hard egret
ooh okay thank you :).
topaz sinewBOT
<@&286206848099549185>
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
@hard egret Has your question been resolved?
topaz sinewBOT
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shell orbit
topaz sinewBOT
shell orbit
Not really sure how to solve this, I feel like it involves trig?
(x+0)^2 + (y+2)^2 = 25, so the midpoint is (0, -2) and the radius is 5
versed cairn
find the tangent line to circle and check if it passes through the point
shell orbit
find the tangent line to circle and check if it passes through the point
How do I do that?
<@&286206848099549185>
granite sundial
formular : (x - h)^2 + (y - k)^2 = r^2
(x1, y1)
(x-h)(x1-h)^2 + (y-k)(y1-k) = r^2
h=0, k=-2, r=5
0+(y+2) * 25 = 25
y=-1
P(0,23) is y = -1
coral estuary
There must be 2 lines
1.set the formula of the line(s)
2.calculate the distance of the point to the line(s)
3.the distance equals the radius of the circle
4.solve the equation of 3.
shell orbit
OK 1 moment I will try it
coral estuary
I am on the way to my dormitory
I'll show you how to do it when I reach my dormitory
shell orbit
I'll show you how to do it when I reach my dormitory
Thank you 🤗
coral estuary
2.calculate the distance of the point to the line(s)
The center of the circle. I mean the point is.
barren zodiac
formula of your line so far is y = ax + 23 (because point (0, 23) applies to this formula)
distance of the circle's center to this line is equal to the circle's radius
you can find "a" by calculating the distance of the circle's center (0, -2) to the line
shell orbit
formula of your line so far is y = ax + 23 (because point (0, 23) applies to thi...
Wait, why is it y = ax + 23? what do you mean by (because point (0, 23) applies to this formula)?
I get that the point is on the line...
barren zodiac
the line that is tangent to the circle passes through P(0 , 23)
say the formula of this line is y = ax + b
if you apply the point P to this formula you'll have:
23 = a(0) + b
so b = 23
coral estuary
If a point If a point is on the line with a slope a.
You can set the formula.
it this question x0 is 0
y0 is 23
shell orbit
What are x0 and y0?
coral estuary
Coordination of a point on the line
shell orbit
formula of your line so far is y = ax + 23 (because point (0, 23) applies to thi...
**you can find "a" by calculating the distance of the circle's center (0, -2) to the line **
25 right?
So thats the slope?
coral estuary
......
You need to solve a and b.
a is the slope
YYou already know the point P is on the line
one point for one unknown number
If you have two points on the line, you can solve a and b at the same time.
But you only have one point
shell orbit
You need to solve a and b.
I thought we had b=23
coral estuary
Of course b is 23 in this question.
because the x0 is 0
but
if x0 is not 0
you can only express b with something about a
barren zodiac
**you can find "a" by calculating the distance of the circle's center (0, -2) to...
the radius of the circle is 5, and your line is tangent to the circle, so the distance between the circle's center and the line is also 5
and you already know b, so you can calculate a by putting the distance between the line and (0, -2) equal to 5
shell orbit
you can only express b with something about a
True, I need to know how to do it without x being equal to 0
coral estuary
x0,y0 is the Coordination of one known point on the line
in this question , x0,y0 is P
And one point can't solve a line
so you need
One known point, and one distance between another known point and the line
tangent line, means the distance is the radius of the circle
one known point, one known distance
so two known conditions
so you can solve a line
luckily, x0 of the point P, is 0
so b is solved directly
if P is (1,23)
shell orbit
the radius of the circle is 5, and your line is tangent to the circle, so the di...
so you can calculate a by putting the distance between the line and (0, -2) equal to 5
Using which formula sry?
shell orbit
tangent line, means the distance is the radius of the circle
Which is 5
coral estuary
Don't you know the distance formula of a points Off the line to the line?
5Is the radius of the circle.
shell orbit
if P is (1,23)
its 0,23 no?
shell orbit
Don't you know the distance formula of a points Off the line to the line?
Im not sure what you mean by off the line to the line
shell orbit
Like the distance between 2 points?
barren zodiac
using the formula they used in this pic, if you have a line with formula
ax + by + c = 0 and coordinates of a point, you can calculate the distance between the point and the line using
d = | a(x0) + b(y0) + c | / sqrt(a^2 + b^2)
coral estuary
using the formula they used in this pic, if you have a line with formula
ax + by...
Yes yes yes yes yes.
nima excellent
barren zodiac
using the formula they used in this pic, if you have a line with formula
ax + by...
(assume coordinates of the point is P(x0, y0) )
shell orbit
But why do we want the distance? That doesnt give us anything in the line
It doesnt give us a or b in y=ax+b
coral estuary
we need
if you want to solve 2 unknown
you need 2 known conditions
but
We only have one known point (0,23)
So
we need the known distance
barren zodiac
It doesnt give us a or b in y=ax+b
since you only have every variable except "a", using that formula can get us to a
shell orbit
You want to solve a and b right?
Yes
coral estuary
and you have P(0,23),right?
it is only one known condition
if you want to solve 2 unknown
you need to find another known condition,right?
coral estuary
Yes
The another condition is that the line is tangent line.
Calculate the distance and you will find another equation you have 2 equations to solve 2 unknown.
shell orbit
Ok so
coral estuary
better read these formulas on math books
shell orbit
|ax1 + by1 + c|/sqrt(a^2 + b^2)
coral estuary
|ax1 + by1 + c|/sqrt(a^2 + b^2)
yes
coral estuary
coral estuary
|ax1 + by1 + c|/sqrt(a^2 + b^2)
It's the distance between x1 y1 and the line ax+ by+ c=0
If you know a point on the line, b can be expressed by a
shell orbit
It's the distance between x1 y1 and the line ax+ by+ c=0
And x1 y1 is the intersection of the line were trying to find and the circle right?
shell orbit
And x1 y1 is the intersection of the line were trying to find and the circle rig...
Its not?
coral estuary
And x1 y1 is the intersection of the line were trying to find and the circle rig...
No
shell orbit
Which point is it
coral estuary
give me some time
shell orbit
Ok
coral estuary
i will explain completely
- you have 2 unknown.
- You need 2 known conditions
- You have one known point on the line,it is (0,23)
4.you need another condition to set another equation
5.you know the line is the tangent line
6.the distance of the center point of the circle to the tangent line is the radius of the circle
coral estuary
|ax1 + by1 + c|/sqrt(a^2 + b^2)
- The distance formule of one point to one line is this
x1y1 is the point, ax + by +c=0 is the line, and dont mix them up
abc are not the ab in the question
a is a by coincidence
but b is not b
b is the coefficience of y
for example
if the line is y=ax+b
it is ax -y +b =0
the a in distance formula is a
the b in distence formula is -1
the c is b
now you have 2 equations
one equation is P applied in the y=ax+b
P(0,23)
the another is
the distance = the radius
apply the center of the circle and the line to the distance formula
you can have the left of the second equation
Look at the formula of the circle
you can have the right of the 2nd equation
coral estuary
you can have the right of the 2nd equation
It is 5
the radius is 5
so
Thats all
👀
shell orbit
1 moment I am reading it and trying to understand
coral estuary
Ok
shell orbit
y=ax-23
y-ax+23
|ax1 + by1 + c|/sqrt(a^2 + b^2)
|(-a)x1 + (-2)y1 + 23|/sqrt((-a)^2 + (1)^2) = 5
M (0, -2)
|(-a)(0) + (-2)(23) + 23|/sqrt((-a)^2 + (1)^2) = 5
So far so good?
shell orbit
Ah yeah sry
barren zodiac
M (0, -2)
you're calculating the distance between the circle's center to the line, so you gotta use the center's coordiantes here
coral estuary
x1y1 is the center
shell orbit
you're calculating the distance between the circle's center to the line, so you ...
Ah ok
shell orbit
|(-a)(0) + (-2)(23) + 23|/sqrt((-a)^2 + (1)^2) = 5
|(0) + (-46) + 23|/sqrt(a^2 + 1) = 5
|-23|/sqrt(a^2 + 1) = 5
coral estuary
Sorry
barren zodiac
i thinnk you made a couple typos
shell orbit
115 = sqrt(a^2 + 1)
coral estuary
I sometimes typo
coral estuary
But
shell orbit
115^2 = a^2 + 1
barren zodiac
|(-a)x1 + (-2)y1 + 23|/sqrt((-a)^2 + (1)^2) = 5
and y1 = -2 here, not 23
barren zodiac
115^2 = a^2 + 1
yes, but not these ones
barren zodiac
Should i get 2 answers for a?
that makes sense because from every point outside of a circle, there are 2 lines that can be tangent to it, so no need to worry about getting 2 answers
barren zodiac
shell orbit
115 = sqrt(a^2 + 1)
Was everything I did correct up to here?
barren zodiac
https://imgur.com/4Qafw1N
sorry for the hideous handwriting, but this is how you apply the formula in this question
coral estuary
yes
shell orbit
sorry for the hideous handwriting, but this is how you apply the formula in this...
Its better than mine haha
shell orbit
5*23
shell orbit
Ah my bad
I got the correct answer tho
I think I know how to do it now
Thank you guys!!
❤️
.close
topaz sinewBOT
barren zodiac
cheers, would recommend pracitcing a bit with the distance formula, it's pretty useful in a lot of questions
shell orbit
cheers, would recommend pracitcing a bit with the distance formula, it's pretty ...
I will 🤗
coral estuary
hahaha
topaz sinewBOT
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compact vector
can someone help me with this one?
topaz sinewBOT
can someone help me with this one?
compact vector
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topaz sinewBOT
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hexed cove
topaz sinewBOT
hexed cove
I could reopen the last channel for some reason
But I reached here from this question
How do I find the value of k and a
There are 3 variables in this equation
<@&286206848099549185>
nocturne flume
There are 3 variables in this equation
its 2 equations but 3 variables?
hexed cove
1
nocturne flume
then in that case theres no 1 singular solution
hexed cove
1 equation 3 variables
nocturne flume
that just means its a multivariable graph
the amount of equations needs to equal the amount of variables for there to be only 1 solution
or not 1 but like a finite set of solutions
coral estuary
hahaha
hexed cove
I divided the p(x) by d(x) and got this
coral estuary
i see it at other channels
hexed cove
i see it at other channels
That one closed
While I was trying to figure out how to solve
hexed cove
Huh
coral estuary
because x
is the variable of poly
two poly equal if and only if
coefficients equal
dont need to care about x
hexed cove
What
coral estuary
just to look at coefficients of x
coral estuary
two poly is the same, when their powers and coefficients equal
because it is POLYNOMIAL
for example
2x+3=ax+3
implies a = 2
for all x
or, it will unequal at many points
hexed cove
Can you tell me the name of theorem
coral estuary
多项式唯一性定理
hexed cove
Also can't polynonials value be different
coral estuary
Fundamental Theorem of Arithmetic
actually this
it is difficult to explain
but you should remember
if two polynomials equal
they are the same thing
otherwise, they just intersect at some points
you can say, the values of them equal, but you cant say they equal
topaz sinewBOT
@hexed cove Has your question been resolved?
rigid cloak
Hlo
rigid cloak
Yo
cedar wagon
Fr
ivory sorrel
Just ask
cedar wagon
About ?
rigid cloak
It should be Calculus
ivory sorrel
Elaborate
cedar wagon
All of the calculus?
rigid cloak
You all choose
We will begin
Calc Analysis and Discussion session
Hlo?
@ivory sorrel
ivory sorrel
I’m kind of working on linear algebra atm sorry
rigid cloak
Which topic?
cedar wagon
Bro, ngl im not into calculus, im on perfectoid spaces actually
rigid cloak
Will linear algebra be good for all?
ivory sorrel
I’m already in a reading group,can’t join another,sorry
rigid cloak
So it's a LINEAR ALGEBRA discussions session
We need 3 bro
2 people talk
3 people discuss
Fr 🔥🗣️
@cedar wagon let's go
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topaz sinewBOT
terse abyss
In calculus anything
A calculus challenge I’ve had on the back of my mind for years is: venn diagrams are made from two intersecting circles, when they overlap there is all intersection and no individual circle area, when they are separate there is no intersection area, so how far apart should the circle centers be to get three equal size areas to make the perfect Venn diagram?
topaz sinewBOT
✅
rigid cloak
Okok i see
terse abyss
I’m not going to help you because it’s hard and a time sink, but if you’re looking for something to do, go for it!
And I’d love to see your solution
shell orbit
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rigid cloak
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topaz sinewBOT
terse abyss
Elaborate a little more
Here
rigid cloak
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topaz sinewBOT
✅
rigid cloak
Oh
I get it
When they intersect all the 3 areas should be same
Okok
I get it
Then u asking how much should the center's be far
Okok
I get it
@yaku
@cedar wagon
Need help
cedar wagon
Yeah
icy sky
say the radius of the circle is 1
so areas are pi
pi/2* whoops
we can split the middle eye shape in half
both vertically and horizontally
to get a curved triangle with area pi/8
then you basically need $\int_a^1 \sqrt{1-x^2}dx=\frac{\pi}{8}$
then find a
thorny flameBOT
Underfull \hbox (badness 10000)
icy sky
what
rigid cloak
U were explaining
icy sky
im done explaining
rigid cloak
√(1-x^2)
I know this integral
x√(1-x^2)/2 +(1/2)arcsinx=π/8
Putting the limits
,w calculate a integral from a to 1 √(1-x^2)=π/8
icy sky
,w fnInt(sqrt(1-x^2),x,t,1)= pi/8
rigid cloak
No bruh
thorny flameBOT
Requested by garlicbredfries
rigid cloak
,w evaluate -(a)(√(1-a^2))/2-(1/2)arcsin(a)=-π/4
@icy sky
The
F
Lol
-1?
Fr
icy sky
the two center need to be approximately 0.8r apart
thorny flameBOT
terse abyss
the two center need to be approximately 0.8r apart
What does that give for areas?
icy sky
(pi/2)r^2 as expected
rigid cloak
Okok
topaz sinewBOT
@rigid cloak Has your question been resolved?
lapis island
topaz sinewBOT
topaz sinewBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
astral blaze
Do you know what is piecewise function?
lapis island
Do you know what is piecewise function?
yeah a little bit
astral blaze
I will hint you that graph looks like 2 lines to me
only if I know what is those two lines and when to choose each one
astral blaze
well... would you agree that the graph looks like two linear functions smash together to make a V?
lapis island
yes
astral blaze
then can I split that into 2 functions, each represent each line?
lapis island
yes
astral blaze
yeah... that is how you solve the problem. You find that line and find the condition of choosing one line over the other
lapis island
@astral blaze can like solve subfunction and domain for a, and tell me how step by step?
only if you want
astral blaze
sure. you know that |x| is just x for x > 0, right?
lapis island
sure. you know that |x| is just x for x > 0, right?
yeah
astral blaze
that means I know that one of my line is f(x) = x for the case of |x|
and that applies for x > 0
lapis island
mhm
astral blaze
so that is one of your subfunction (for the case of |x|)
another line is just the case of x <= 0
Here, in your problem, you are dealing with |x-3|
use the same principle and you should be able to do that. Again, notice that for x <= 0, |x| = -x (why?)
topaz sinewBOT
@lapis island Has your question been resolved?
lapis island
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topaz sinewBOT
topaz sinewBOT
Available help channel!
cyan echo
Question: Given that ABCD is a quadrilateral, E and F are midpoints of the sides AD and BC of ABCD. Suppose that AB and CD are not parallel, prove that 2EF < AB + CD.
dont rlly know how to approach this question
topaz sinewBOT
Question: Given that ABCD is a quadrilateral, E and F are midpoints of the sides...
rigid ivy
!occupied
topaz sinewBOT
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
cyan echo
Question: Given that ABCD is a quadrilateral, E and F are midpoints of the sides...
<@&286206848099549185>
topaz sinewBOT
@cyan echo Has your question been resolved?
cyan echo
<@&286206848099549185>
granite sundial
median pivot
He said (AB) and (CD) are not parallel not (AD) and (BC)
granite sundial
He said (AB) and (CD) are not parallel not (AD) and (BC)
got it
I jsut try again
granite sundial
<@&286206848099549185>
Did you learn about vector?
topaz sinewBOT
@cyan echo Has your question been resolved?
cyan echo
<@&286206848099549185>
granite sundial
granite sundial
how do yk ag = 2ef tho
Do you understand?
granite sundial
DC//BG
limpid quest
hey i need help with this adding fraction problem.
topaz sinewBOT
hey i need help with this adding fraction problem.
limpid quest
1/2 + 7/26 + 3/32 + 3/8
so i used 32 as the main denominator and added the numerators and got 14. is the answer 14/32?
dense lily
so i used 32 as the main denominator and added the numerators and got 14. is the...
32??
dense lily
yes bc of 3/32
No. You're meant to find the lowest common multiple of the denominators
That'll be the denominator
limpid quest
No. You're meant to find the lowest common multiple of the denominators
how can i find it
glass canyon
also useful trick, the least common multiple of 4 numbers can be simplified from $\text{lcm}(a,b,c,d,\dots)$ to $\text{lcm}(\lcm(a,b,c),d)$.
dense lily
how can i find it
I will just send a vi
glass canyon
and since 2,32, and 8 are divisible, lcm(2,32,8) is 32.
thorny flameBOT
fish
Compile Error! Click the 
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limpid quest
so it would be 416?
topaz sinewBOT
@limpid quest Has your question been resolved?
analog dome
this question has been driving me crazy
topaz sinewBOT
this question has been driving me crazy
analog dome
how am i supposed to make a function that looks similar to that
rigid ivy
Looks like a sine function
analog dome
it does, but
how would iget it to be like that
since sine is usually just neat with the same max and min
jade thunder
How would you need to change sine to make it look like that?
That’s true, how could you change that?
What else does it look like?
Besides the wavy sine part
analog dome
it looks like its symmetric about the origin. it also looks like the same wave repeated over and over again, but downward?
like its the same pattern
analog dome
a decreasing function?
jade thunder
Which one
analog dome
negative cubic?
analog dome
im not sure
jade thunder
Uh, y = -x
analog dome
oh lol
jade thunder
Yeah
That graph is just the amalgamation of these 2 functions
If I had to guess
It’s a sine wave along the y = -x line
Or, y = sinx + (-x)
analog dome
this is insane
jade thunder
It’s clever!
analog dome
ive been dealing with trig graphs for years, but
i didnt know that its the amalgamation of the 2 functions
thats why sinx + 1 goes along x = 1
wow
this is interesting
jade thunder
Yep!
analog dome
thanks!
jade thunder
thats why sinx + 1 goes along x = 1
That is exactly the intuition you should have here
analog dome
mhm
have a great day, man
such an interesting fact
i am now satisfied
now*
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topaz sinewBOT
topaz sinewBOT
Available help channel!
granite stag
So i need help understanding WHY the period of this trig problem is just pi and how to calculate that, if anyone could explain it to me that would be very helpful as thats the only part i am stuck on. to save you time the final answer is y = 6cos2x
topaz sinewBOT
So i need help understanding WHY the period of this trig problem is just pi and ...
craggy haven
it's a little hard to see but if you look at the horizontal distance between two peaks it's π
it might be easier to see the horizontal distance between a peak and a trough, which is π/2
granite stag
it's a little hard to see but if you look at the horizontal distance between two...
I apologize I cant make the image clearer
craggy haven
no i mean it's hard to see because of how they cut the image off
granite stag
ahhh, yeah every problem is like that in this class sigh
craggy haven
you can use the half-period then, by looking at the (horizontal) distance between the peak and the trough
granite stag
could it be that cosx from what ive noticed has always been on a pi period?
so it would always just be pi if its cosx? or no
craggy haven
cosine has a period of 2π
however, this is cos(2x) which is horizontally squished, so it has a period of π
granite stag
Im confused, so would I still have to use the formula P = 2pi/b? to find this period
craggy haven
how does that formula work
granite stag
Not sure, been struggling on this unit alot and while i know how to use the formula I dont know exactly how it works
craggy haven
okay, how do you use the formula?
like what pattern does the function have to have for it to work
eg what does b mean?
craggy haven
if you have the graph, you can see the period by just measuring it
granite stag
measuring from which point?
craggy haven
peak to peak, or trough to trough
or really anywhere that's "the same"
like in this case, it dips below the x axis twice
you can measure from there, and those are a distance of π apart
granite stag
Im trying to make sense of this i apologize if im not understanding as quick
So if i measure from the peak which is 6 for both, the distance is pi?
craggy haven
okay it says wavelength but that's basically the same as period, just one is in distance and one is in time
craggy haven
So if i measure from the peak which is 6 for both, the distance is pi?
yes
do you understand why the distance is π?
granite stag
do you understand why the distance is π?
I do not
craggy haven
look at the markings on the horizontal axis
granite stag
the pi/2, pi and 3pi/2
craggy haven
this distance is π
craggy haven
this distance is also π
craggy haven
what's more, this distance is also π
and if you were to extend the wave in both directions, you'd see that the function repeats itself over and over again, at a distance of π each time
topaz sinewBOT
@granite stag Has your question been resolved?
spring tide
topaz sinewBOT
spring tide
Plz check my work
ivory wave
how did you get 9 from
spring tide
which question?
spring tide
Cuz the 6 is include in the bracket
ivory wave
you wrote ¾f
ignore the ¾ I thought my keyboard has 3/2
but anyway it's wrong
I see the question has 2/3
ivory wave
I'm confused 
ivory wave
Multiply 3/2 both side
it says 2/3y in the question I don't get where are you getting 3/2 from